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Example Real-World Application Pulsed NMR is the most common approach to doing modern-day NMR. The NMR technique developed by Purcell and Bloch in 1946 used constant wave NMR where electromagnetic radiation was continuously applied to the sample and then the frequency was steadily changed to build up frequency absorption spectrum. Pulsed NMR was developed by Erwin Hahn in 1950 and made use of short-duration electromagnetic pulses to excite a wide range of nuclei all at once. The acquired free induction decay signal is ideal for Fourier analysis to quickly visualize the different resonant frequencies present. Over time, more advanced NMR pulse sequences were developed that enable techniques like signal enhancement, separating out different forms of NMR interactions contributing to the signal, multidimensional NMR spectroscopy, and MR imaging.
At the end of this module, students should be able to…
Use Bloch simulator and a physical model to answer questions about the spin dynamics resulting from a given pulse sequence (Scientific Ability A4)
Extract information from provided NMR experimental data (Scientific Ability A1)
Identify a hypothesis that can be tested for determining the source of a mysterious signal (Scientific Ability C1)
“There is nothing that nuclear spins will not do for you, as long as you treat them as human beings.”
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Erwin L. Hahn - often referred to as the “Wizard of Magnetic Resonance” for developing the use of pulsed NMR and his ‘accidental discovery’ of the spin echo. Along with his many scientific contributions, Hahn was well-known for his humor and talents as a musician and entertainer (2).
“Good morning! I feel a bit like a mosquito at a beach…” and indicated the numerous props to the startled audience. “I don’t know where to start!” - Erwin Hahn, when presenting a physics of music lecture In order to determine the \(T_1\) relaxation time in the previous module, your experiment most likely involved multiple pulses. In fact, very soon after coming up with using an electromagnetic pulse and measuring the free induction decay, Erwin Hahn discovered an fascinating phenomenon when he looked at the NMR signal after multiple pulses. Hahn was fond of telling the story of his ‘accidental discovery’ which occurred he was a post doc at the University of Illinois, Urbana. Similar to the application experiment in the previous module, Hahn was measuring nuclear relaxation times when he applied two pulses separated by a time interval instead of one. Along with the expected FID signal after each 90\(^\circ\) pulse, he saw a puzzling ‘ghost’ signal following the second pulse that he initially thought must have been the result of a glitch. Hahn’s later realization of the cause of this puzzling signal - which he termed a ‘spin echo’ (3) - ultimately led to the development multiple pulse sequences that enable a diverse amount of modern-day MR techniques.
In this module, you will undergo your own explorations of spin echoes making use of visualizing the spin dynamics on the Bloch sphere and looking at the results of various NMR experiments. Let’s begin with a review of interpreting pulse sequence diagrams by analyzing the pulse sequence shown below.
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What state is the net nuclear magnetization vector in at the end of the relaxation delay period? Right after the inversion (\(\pi\)) pulse? At the end of the delay period (\(\tau\))? Right after the second \(\pi/2\) (i.e. 90\(^\circ\)) pulse?
This is called an inversion recovery pulse sequence. Why do you think it is called that? What relaxation time might this experiment be helpful to measure?
Let’s return, one more time, to the Bloch Simulator to see if we can generate our own spin echoes and make sense of how the MR signal can potentially be coming back.
Set up the Bloch simulator to use ‘Inhomogeneity’(and option in the ‘Equilibrium’ menu) so that we will see multiple spins responding to an inhomogeneous external magnetic field. We can leave relaxation off for now to make any echo appear more obvious. Let’s hop into the rotating frame (set frame to ‘B0’) to also help clarify what we see.
Fig. 34 from MRI Made Easy by Prof. Dr. Hans H. Schild (5). One possible physical analogy for the Hahn spin echo.
After setting up the simulator as described above, knock-down the spins with a hard-\(90^\circ_x\) pulse and draw a sketch and write a description of what you see. Add some arrows to your sketch showing which spins are precessing clockwise and which are precessing counterclockwise in the rotating frame. Recalling how we set up the simulation, what is causing the spins to dephase from each other? What relaxation time would characterize the resulting MR signal decay?
Start the spins at equilibrium again (by clicking on ‘Inhomogeneity’), knock-down the spins with a hard-\(90^\circ_x\) pulse and after some time, apply a \(180^\circ_y\) pulse. Draw a sketch and write a description of what you see after the \(180^\circ_y\) pulse is applied. Keep track of the the direction of the spins precession in the rotating frame before and after the \(180^\circ\) pulse. Does the direction of the each individual spin’s precession change with the pulse? Does this make sense considering the direction of the external magnetic field has not changed?
It is helpful to have a physical model in your head to make sense of the spin dynamics on the Bloch sphere that lead to spin echoes - some favorites are racers on a race track or opening/closing a folding fan. Choose your favorite physical model and explain what causes the echo you observe, in your own words.
FUN FACT! You can get spin echoes with any two pulses, but different spins can refocus at different times. The traditional Hahn echo pulse sequence (\(90^\circ\) pulse followed by \(180^\circ\) pulse) optimizes the echo so that all spins precessing at slightly different frequencies refocus at the same time. You can read more at the following link, which includes an explanation of what Hahn referred to as the “eight-ball echo”: https://www.mriquestions.com/90deg-90deg-hahn-echo.html.
Does the phase of the pulses (that is, whether they are applied in the x- or y- direction) appear to determine whether you see an echo or not? Apply different combinations of pulses on the simulator and your physical model to settle on your answer.
If the time between the \(90^\circ\) and \(180^\circ\) pulses is \(\tau\), how long after the \(180^\circ\) pulse do you expect to see the peak of the echo? Use the simulator and your physical model to settle on your answer.
coherence- when the quantum states of the system stay correlated with each other; in quantum computing the coherence time relates to how long experimenters should expect the qubit to retain its current quantum state before it relaxes due to interactions with its environment
The traditional Hahn echo pulse sequence (\(90^\circ\) pulse followed by \(180^\circ\) pulse) can refocus any dephasing along the transverse plane due to external magnetic field inhomogeneities because the \(180^\circ\) pulse has the effect of flipping the spins with respect to the external magnetic field. Each individual spin is still precessing with the same frequency and direction as before, but effectively transplanted to a new spot on the Bloch sphere due to the \(180^\circ\) pulse. The happen to be transplanted to the exact spot on the block sphere so that, as time goes on, they completely reverse whatever dephasing had previously occurred prior to the \(180^\circ\) pulse. This appeared to be an apparent time reversal of what had been considered by everyone at the time a irreversible process! In the thermodynamics sense, whatever is mixed will not naturally become unmixed, even if you try to now mix it in the opposite direction. Hahn showed that there remained some hidden order in the MR signal (usually given the fancy name of coherence) that could be restored through the clever use of pulses.
Gavin W Morley, CC
BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via
Wikipedia. You can find more information on the file
information page.
The Hahn echo provides a way to effectively get rid of the effect on the spins due to external magnetic field inhomogeneities - the primary cause of the short \(T_2^*\) relaxation time constant. However, since presumably all spins are being flipped by \(180^\circ\), this pulse will not impact the spin-spin interactions - since two magnets will still have the same interaction between them when you flip both of their orientations). The remaining decay of the Hahn echo signal must then be primarily due to spin-spin interactions.
Gavin W Morley, CC
BY-SA 3.0 https://creativecommons.org/licenses/by-sa/3.0, via
Wikimedia Commons. You can find more information on the file
information page.
What relaxation time constant should the Hahn echo experiment enable you to measure?
Describe an experimental procedure that you could use to measure this relaxation time constant.
Being able to measure the \(T_2\) relaxation time is important for characterizing our NMR samples. Below is the multiple-pulse sequence developed by Carr, Purcell, Meiboom, and Gill (CPMG) that is the primary method of measuring the \(T_2\) relaxation time.
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How does the CPMG pulse sequence compare with the experimental procedure you developed in the previous question?
TE is the shorthand for the ‘echo time’ or the time spacing between consecutive 180\(^\circ\) pulses. Why does it make sense that the time between the initial 90\(^\circ\) pulse and the first 180\(^\circ\) pulse is TE/2?
What are some advantages to using the CPMG pulse sequence instead of just the standard Hahn echo pulse sequence to measure \(T_2\)?
Describe how you would go about determining the \(T_2\) relaxation time constant for a sample if given data from a CPMG experiment.
Below is some \(^1\)H CPMG data
collected using a heavy mineral oil sample. What do you think is plotted
along the y- and x-axes? Estimate the \(T_2\) relaxation time constant.
Below is some \(^1\)H CPMG data
collected from a neoprene sample. Estimate the \(T_2\) relaxation time constant.
From the \(T_2\) values you found above, what can you say about the local magnetic environment of neoprene as compared with mineral oil (e.g. is it more or less homogeneous)?
Check out the link in the FUN FACT! text found in the margin
above to help answer the following questions about the mysterious NMR
signal below that was collected repeating an FID experiment with a short
TR time.
Using whichever representation or model you prefer, explain a possible source for the small signal we see right before each subsequent \(90^\circ\) pulse. Why does it make sense that it appears to peak at the time the next pulse occurs?
Provide a pulse sequence and an experimental procedure to test your hypothesis for the source of the signal.
